Compact non-orientable surfaces of genus 4 with extremal metric discs

Abstract

Abstract. A compact hyperbolic surface of genus g is said to be extremal if it admits an extremal disc, a disc of the largest radius determined by g. We know how many extremal discs are embedded in a non-orientable extremal surface of genus g = 3 or g > 6. We show in the present paper that there exist 144 non-orientable extremal surfaces of genus 4, and find the locations of all extremal discs in those surfaces. As a result, each surface contains at most two extremal discs. Our methods used here are similar to those in the case of g = 3.

A brute force computer aided proof of an existence result about extremal hyperbolic surfaces

“…Notice that four of the eleven required surfaces can be found already in the literature, since they correspond to the cases for which k N = 1 (see [GN07], [Nak09], [Nak12], [Nak13] and [Nak16]). These surfaces correspond to N = 12, 18, 24 or 30.…”

Compact non-orientable surfaces of genus 6 with extremal metric discs

“…The third column of Table 1 shows the group of automorphisms Aut ± (S j ) of S j in the category of Klein surfaces. As a result of Theorem 2.1 and the results of g 3 (g = 6) ( [6], [9], [10], [11]) we have the following:…”

“…For the orientable case, the problem is completely solved ( [4], [5], [7], [8], [12]). For the non-orientable case, we know that extremal surfaces of genus g > 6 admit a unique extremal disc ( [6]) and that those of genus 3, 4 or 5 admit at most two ( [6], [9], [10], [11]). Those surfaces are presented by fundamental regions of their Fuchsian groups or non-Euclidean crystallographic groups (NEC groups).…”

Extremal k-packings in compact non-orientable surfaces

“…For example, for k = 1 Theorem 8 says that 1-packings in compact non-orientable surfaces will be unique if 6g − 6 is not in the above list of numbers, which precisely occurs for g > 6 (consistent with [GN07]). In the series of papers [GN07], [Nak09], [Nak12], [Nak13] and [Nak16] all 1-extremal surfaces of genus g from 3 to 6 were studied in detail, and in particular the existence of extremal surfaces with more than one extremal disc was explicitly shown for these values of g. Now, suppose that there exists a compact non-orientable primitive k N -extremal surface X of genus g N with multiple extremal k N -packings. Denote N = 6g N + 6k N − 12 k .…”